Let M be an (mx n)-matrix with nonnegative real entries and let r, 12,...,m denote the rows of M. Assume that ker(M) nZ" = {0}. Define a binary relation >M on the monomials in the polynomial ring S = K[x1,x2,...,xn] as follows: x">M x if there is an positive integer i (at most m) such that u r;= v r; for all 1 M is a monomial order on the polynomial ring S. (ii) When M:=ii [], show that >M equals > grevlex on K[x, y, z). (iii) For the (nx n)-identity matrix I, show that >lex equals >1.